By part (3) of the main result of [1]}, we know that there is some nonzero form \(f\in S_1(\Gamma _0(N), \mathbb {F}_2)^{\operatorname{Katz}}\)  with \(f^2=g\) . (Here, we're considering weight 1 Katz forms, and so the weight 2 forms we construct may be Katz forms as well. So if necessary we enlarge the spaces we're considering, but it doesn't affect the conclusion.) As forms with coefficients in \(\mathbb {F}_2\)  commute with the Frobenius endomorphism, \(f(q^2)\)  has the same \(q\) -expansion as \(g\) . If 1 and \({1,\operatorname{an}}\)  are the weight 1 Hecke algebras, it is quick to check that the corresponding Hecke actions on \(q\) -expansions of \({1,\operatorname{an}}\)  are identical to those of \({\operatorname{an}}\) . Therefore \(f\in S_1(\Gamma _0(N), \mathbb {F}_2)^{\operatorname{Katz}}[\mathfrak {m}]\) . Further, we know that \(f\)  is alone in this space, by part (2) of [1]}: any other form in \(S_1(\Gamma _0(N), \mathbb {F}_2)^{\operatorname{Katz}}[\mathfrak {m}]\)  has the same odd coefficients, so the difference between it and \(f\)  has only even-power coefficients, and hence must be 0 by Katz's theorem. So \(f\)  is also an eigenform for \(T_2\)  in weight 1, say with eigenvalue \(b_2\) .
